Abstract

Several formulations of the differential theory for anisotropic gratings are investigated numerically. Conventional formulations and recent formulations based on Li’s Fourier factorization rules are applied to a sinusoidal-profiled grating made of an anisotropic and conducting material. For both types of formulation, the numerical results of the differential and the rigorous coupled-wave methods are presented, and only the differential method based on Li’s Fourier factorization rules provides a reliable convergence. Moreover, several numerical integration schemes used on the differential method are examined, and the advantage of the implicit integration schemes is shown.

© 2002 Optical Society of America

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References

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  1. R. Petit, “Diffraction d’une onde plane par un réseau métallique,” Rev. Opt. 45, 353–370 (1966).
  2. G. Cerutti-Maori, R. Petit, M. Cadilhac, “Etude numérique du champ diffracté par un réseau,” C. R. Acad. Sci. 268, 1060–1063 (1969).
  3. P. Vincent, “Differential methods,” in Electromagnetic Theory of Gratings, Vol. 22 of Topics in Current Physics, R. Petit, ed. (Springer-Verlag, Berlin, 1980), Chap. 4.
    [CrossRef]
  4. S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 23, 123–133 (1975).
    [CrossRef]
  5. M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 71, 811–818 (1977).
    [CrossRef]
  6. M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. 72, 1385–1392 (1982).
    [CrossRef]
  7. M. Nevière, “Bragg–Fresnel multilayer gratings: electromagnetic theory,” J. Opt. Soc. Am. A 11, 1835–1845 (1994).
    [CrossRef]
  8. M. C. Hutley, J. P. Verrill, R. C. McPhedran, M. Nevière, P. Vincent, “Presentation and verification of a differential formulation for the diffraction by conducting gratings,” Nouv. Rev. Opt. 6, 87–95 (1975).
    [CrossRef]
  9. M. Nevière, R. Petit, M. Cadilhac, “About the theory of optical grating coupler-waveguide systems,” Opt. Commun. 8, 113–117 (1973).
    [CrossRef]
  10. M. Nevière, P. Vincent, R. Petit, M. Cadilhac, “Systematic study of resonances of holographic thin film couplers,” Opt. Commun. 9, 48–53 (1973).
    [CrossRef]
  11. G. Tayeb, “Contribution à l’etude de la diffraction des ondes électromagnétiques par des réseaux. Réflexions sur les méthodes existantes et sur leur extension aux milieux anisotropes,” Ph.D. dissertation, No. 90/Aix 3/0065 (University of Aix-Marseille, Marseille, France, 1990).
  12. M. Nevière, P. Vincent, R. Petit, “Sur la théorie du réseau conducteur et ses applications à l’optique,” Nouv. Rev. Opt. 5, 65–77 (1974).
  13. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996).
    [CrossRef]
  14. E. Popov, M. Nevière, “Grating theory: new equations in Fourier space leading to fast converging results for TM polarization,” J. Opt. Soc. Am. A 17, 1773–1784 (2000).
    [CrossRef]
  15. K. Watanabe, R. Petit, M. Nevière, “Differential theory of gratings made of anisotropic materials,” J. Opt. Soc. Am. A 19, 325–334 (2002).
    [CrossRef]
  16. K. Watanabe, K. Yasumoto, “Reformulation of differential method for anisotropic gratings in conical mounting,” in Proceedings of the Eighth International Symposium on Microwave and Optical Technology (Polytechnic International, Montreal, Canada, 2001), 443–446.
  17. E. Popov, M. Nevière, “Maxwell equations in Fourier space: fast-converging formulation for diffraction by arbitrary shaped, periodic, anisotropic media,” J. Opt. Soc. Am. A 18, 2886–2894 (2001).
    [CrossRef]
  18. K. Watanabe, “Fast converging formulation of differential theory for non-smooth gratings made of anisotropic materials,” Radio Sci. (to be published).
  19. L. Li, “Reformulation of the Fourier modal method for surface-relief grating made with anisotropic materials,” J. Mod. Opt. 45, 1313–1334 (1998).
    [CrossRef]
  20. E. Popov, M. Nevière, B. Gralak, G. Tayeb, “Staircase approximation validity for arbitrary shaped gratings,” J. Opt. Soc. Am. A 19, 33–42 (2002).
    [CrossRef]
  21. Actually, the distinction between upward and downward waves is not obvious in anisotropic media. A detail discussion is given in Refs. 11and 19.
  22. L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A 13, 1024–1035 (1996).
    [CrossRef]
  23. F. Montiel, M. Nevière, P. Peyrot, “Waveguide confinement of Cerenkov second-harmonic generation through a graded-index grating coupler: electromagnetic optimization,” J. Mod. Opt. 45, 2169–2186 (1998).
    [CrossRef]

2002 (2)

2001 (1)

2000 (1)

1998 (2)

L. Li, “Reformulation of the Fourier modal method for surface-relief grating made with anisotropic materials,” J. Mod. Opt. 45, 1313–1334 (1998).
[CrossRef]

F. Montiel, M. Nevière, P. Peyrot, “Waveguide confinement of Cerenkov second-harmonic generation through a graded-index grating coupler: electromagnetic optimization,” J. Mod. Opt. 45, 2169–2186 (1998).
[CrossRef]

1996 (2)

1994 (1)

1982 (1)

1977 (1)

1975 (2)

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 23, 123–133 (1975).
[CrossRef]

M. C. Hutley, J. P. Verrill, R. C. McPhedran, M. Nevière, P. Vincent, “Presentation and verification of a differential formulation for the diffraction by conducting gratings,” Nouv. Rev. Opt. 6, 87–95 (1975).
[CrossRef]

1974 (1)

M. Nevière, P. Vincent, R. Petit, “Sur la théorie du réseau conducteur et ses applications à l’optique,” Nouv. Rev. Opt. 5, 65–77 (1974).

1973 (2)

M. Nevière, R. Petit, M. Cadilhac, “About the theory of optical grating coupler-waveguide systems,” Opt. Commun. 8, 113–117 (1973).
[CrossRef]

M. Nevière, P. Vincent, R. Petit, M. Cadilhac, “Systematic study of resonances of holographic thin film couplers,” Opt. Commun. 9, 48–53 (1973).
[CrossRef]

1969 (1)

G. Cerutti-Maori, R. Petit, M. Cadilhac, “Etude numérique du champ diffracté par un réseau,” C. R. Acad. Sci. 268, 1060–1063 (1969).

1966 (1)

R. Petit, “Diffraction d’une onde plane par un réseau métallique,” Rev. Opt. 45, 353–370 (1966).

Bertoni, H. L.

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 23, 123–133 (1975).
[CrossRef]

Cadilhac, M.

M. Nevière, R. Petit, M. Cadilhac, “About the theory of optical grating coupler-waveguide systems,” Opt. Commun. 8, 113–117 (1973).
[CrossRef]

M. Nevière, P. Vincent, R. Petit, M. Cadilhac, “Systematic study of resonances of holographic thin film couplers,” Opt. Commun. 9, 48–53 (1973).
[CrossRef]

G. Cerutti-Maori, R. Petit, M. Cadilhac, “Etude numérique du champ diffracté par un réseau,” C. R. Acad. Sci. 268, 1060–1063 (1969).

Cerutti-Maori, G.

G. Cerutti-Maori, R. Petit, M. Cadilhac, “Etude numérique du champ diffracté par un réseau,” C. R. Acad. Sci. 268, 1060–1063 (1969).

Gaylord, T. K.

Gralak, B.

Hutley, M. C.

M. C. Hutley, J. P. Verrill, R. C. McPhedran, M. Nevière, P. Vincent, “Presentation and verification of a differential formulation for the diffraction by conducting gratings,” Nouv. Rev. Opt. 6, 87–95 (1975).
[CrossRef]

Li, L.

McPhedran, R. C.

M. C. Hutley, J. P. Verrill, R. C. McPhedran, M. Nevière, P. Vincent, “Presentation and verification of a differential formulation for the diffraction by conducting gratings,” Nouv. Rev. Opt. 6, 87–95 (1975).
[CrossRef]

Moharam, M. G.

Montiel, F.

F. Montiel, M. Nevière, P. Peyrot, “Waveguide confinement of Cerenkov second-harmonic generation through a graded-index grating coupler: electromagnetic optimization,” J. Mod. Opt. 45, 2169–2186 (1998).
[CrossRef]

Nevière, M.

K. Watanabe, R. Petit, M. Nevière, “Differential theory of gratings made of anisotropic materials,” J. Opt. Soc. Am. A 19, 325–334 (2002).
[CrossRef]

E. Popov, M. Nevière, B. Gralak, G. Tayeb, “Staircase approximation validity for arbitrary shaped gratings,” J. Opt. Soc. Am. A 19, 33–42 (2002).
[CrossRef]

E. Popov, M. Nevière, “Maxwell equations in Fourier space: fast-converging formulation for diffraction by arbitrary shaped, periodic, anisotropic media,” J. Opt. Soc. Am. A 18, 2886–2894 (2001).
[CrossRef]

E. Popov, M. Nevière, “Grating theory: new equations in Fourier space leading to fast converging results for TM polarization,” J. Opt. Soc. Am. A 17, 1773–1784 (2000).
[CrossRef]

F. Montiel, M. Nevière, P. Peyrot, “Waveguide confinement of Cerenkov second-harmonic generation through a graded-index grating coupler: electromagnetic optimization,” J. Mod. Opt. 45, 2169–2186 (1998).
[CrossRef]

M. Nevière, “Bragg–Fresnel multilayer gratings: electromagnetic theory,” J. Opt. Soc. Am. A 11, 1835–1845 (1994).
[CrossRef]

M. C. Hutley, J. P. Verrill, R. C. McPhedran, M. Nevière, P. Vincent, “Presentation and verification of a differential formulation for the diffraction by conducting gratings,” Nouv. Rev. Opt. 6, 87–95 (1975).
[CrossRef]

M. Nevière, P. Vincent, R. Petit, “Sur la théorie du réseau conducteur et ses applications à l’optique,” Nouv. Rev. Opt. 5, 65–77 (1974).

M. Nevière, P. Vincent, R. Petit, M. Cadilhac, “Systematic study of resonances of holographic thin film couplers,” Opt. Commun. 9, 48–53 (1973).
[CrossRef]

M. Nevière, R. Petit, M. Cadilhac, “About the theory of optical grating coupler-waveguide systems,” Opt. Commun. 8, 113–117 (1973).
[CrossRef]

Peng, S. T.

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 23, 123–133 (1975).
[CrossRef]

Petit, R.

K. Watanabe, R. Petit, M. Nevière, “Differential theory of gratings made of anisotropic materials,” J. Opt. Soc. Am. A 19, 325–334 (2002).
[CrossRef]

M. Nevière, P. Vincent, R. Petit, “Sur la théorie du réseau conducteur et ses applications à l’optique,” Nouv. Rev. Opt. 5, 65–77 (1974).

M. Nevière, R. Petit, M. Cadilhac, “About the theory of optical grating coupler-waveguide systems,” Opt. Commun. 8, 113–117 (1973).
[CrossRef]

M. Nevière, P. Vincent, R. Petit, M. Cadilhac, “Systematic study of resonances of holographic thin film couplers,” Opt. Commun. 9, 48–53 (1973).
[CrossRef]

G. Cerutti-Maori, R. Petit, M. Cadilhac, “Etude numérique du champ diffracté par un réseau,” C. R. Acad. Sci. 268, 1060–1063 (1969).

R. Petit, “Diffraction d’une onde plane par un réseau métallique,” Rev. Opt. 45, 353–370 (1966).

Peyrot, P.

F. Montiel, M. Nevière, P. Peyrot, “Waveguide confinement of Cerenkov second-harmonic generation through a graded-index grating coupler: electromagnetic optimization,” J. Mod. Opt. 45, 2169–2186 (1998).
[CrossRef]

Popov, E.

Tamir, T.

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 23, 123–133 (1975).
[CrossRef]

Tayeb, G.

E. Popov, M. Nevière, B. Gralak, G. Tayeb, “Staircase approximation validity for arbitrary shaped gratings,” J. Opt. Soc. Am. A 19, 33–42 (2002).
[CrossRef]

G. Tayeb, “Contribution à l’etude de la diffraction des ondes électromagnétiques par des réseaux. Réflexions sur les méthodes existantes et sur leur extension aux milieux anisotropes,” Ph.D. dissertation, No. 90/Aix 3/0065 (University of Aix-Marseille, Marseille, France, 1990).

Verrill, J. P.

M. C. Hutley, J. P. Verrill, R. C. McPhedran, M. Nevière, P. Vincent, “Presentation and verification of a differential formulation for the diffraction by conducting gratings,” Nouv. Rev. Opt. 6, 87–95 (1975).
[CrossRef]

Vincent, P.

M. C. Hutley, J. P. Verrill, R. C. McPhedran, M. Nevière, P. Vincent, “Presentation and verification of a differential formulation for the diffraction by conducting gratings,” Nouv. Rev. Opt. 6, 87–95 (1975).
[CrossRef]

M. Nevière, P. Vincent, R. Petit, “Sur la théorie du réseau conducteur et ses applications à l’optique,” Nouv. Rev. Opt. 5, 65–77 (1974).

M. Nevière, P. Vincent, R. Petit, M. Cadilhac, “Systematic study of resonances of holographic thin film couplers,” Opt. Commun. 9, 48–53 (1973).
[CrossRef]

P. Vincent, “Differential methods,” in Electromagnetic Theory of Gratings, Vol. 22 of Topics in Current Physics, R. Petit, ed. (Springer-Verlag, Berlin, 1980), Chap. 4.
[CrossRef]

Watanabe, K.

K. Watanabe, R. Petit, M. Nevière, “Differential theory of gratings made of anisotropic materials,” J. Opt. Soc. Am. A 19, 325–334 (2002).
[CrossRef]

K. Watanabe, “Fast converging formulation of differential theory for non-smooth gratings made of anisotropic materials,” Radio Sci. (to be published).

K. Watanabe, K. Yasumoto, “Reformulation of differential method for anisotropic gratings in conical mounting,” in Proceedings of the Eighth International Symposium on Microwave and Optical Technology (Polytechnic International, Montreal, Canada, 2001), 443–446.

Yasumoto, K.

K. Watanabe, K. Yasumoto, “Reformulation of differential method for anisotropic gratings in conical mounting,” in Proceedings of the Eighth International Symposium on Microwave and Optical Technology (Polytechnic International, Montreal, Canada, 2001), 443–446.

C. R. Acad. Sci. (1)

G. Cerutti-Maori, R. Petit, M. Cadilhac, “Etude numérique du champ diffracté par un réseau,” C. R. Acad. Sci. 268, 1060–1063 (1969).

IEEE Trans. Microwave Theory Tech. (1)

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 23, 123–133 (1975).
[CrossRef]

J. Mod. Opt. (2)

L. Li, “Reformulation of the Fourier modal method for surface-relief grating made with anisotropic materials,” J. Mod. Opt. 45, 1313–1334 (1998).
[CrossRef]

F. Montiel, M. Nevière, P. Peyrot, “Waveguide confinement of Cerenkov second-harmonic generation through a graded-index grating coupler: electromagnetic optimization,” J. Mod. Opt. 45, 2169–2186 (1998).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (7)

Nouv. Rev. Opt. (2)

M. Nevière, P. Vincent, R. Petit, “Sur la théorie du réseau conducteur et ses applications à l’optique,” Nouv. Rev. Opt. 5, 65–77 (1974).

M. C. Hutley, J. P. Verrill, R. C. McPhedran, M. Nevière, P. Vincent, “Presentation and verification of a differential formulation for the diffraction by conducting gratings,” Nouv. Rev. Opt. 6, 87–95 (1975).
[CrossRef]

Opt. Commun. (2)

M. Nevière, R. Petit, M. Cadilhac, “About the theory of optical grating coupler-waveguide systems,” Opt. Commun. 8, 113–117 (1973).
[CrossRef]

M. Nevière, P. Vincent, R. Petit, M. Cadilhac, “Systematic study of resonances of holographic thin film couplers,” Opt. Commun. 9, 48–53 (1973).
[CrossRef]

Rev. Opt. (1)

R. Petit, “Diffraction d’une onde plane par un réseau métallique,” Rev. Opt. 45, 353–370 (1966).

Other (5)

P. Vincent, “Differential methods,” in Electromagnetic Theory of Gratings, Vol. 22 of Topics in Current Physics, R. Petit, ed. (Springer-Verlag, Berlin, 1980), Chap. 4.
[CrossRef]

K. Watanabe, K. Yasumoto, “Reformulation of differential method for anisotropic gratings in conical mounting,” in Proceedings of the Eighth International Symposium on Microwave and Optical Technology (Polytechnic International, Montreal, Canada, 2001), 443–446.

G. Tayeb, “Contribution à l’etude de la diffraction des ondes électromagnétiques par des réseaux. Réflexions sur les méthodes existantes et sur leur extension aux milieux anisotropes,” Ph.D. dissertation, No. 90/Aix 3/0065 (University of Aix-Marseille, Marseille, France, 1990).

K. Watanabe, “Fast converging formulation of differential theory for non-smooth gratings made of anisotropic materials,” Radio Sci. (to be published).

Actually, the distinction between upward and downward waves is not obvious in anisotropic media. A detail discussion is given in Refs. 11and 19.

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Figures (6)

Fig. 1
Fig. 1

Geometry of the grating under consideration.

Fig. 2
Fig. 2

Layers used in the S-matrix propagation algorithm.

Fig. 3
Fig. 3

Staircase approximation for the rigorous coupled-wave method.

Fig. 4
Fig. 4

Comparison of the convergences of diffraction efficiencies computed by various formulations (Old, conventional formulations; New, recent formulations based on Li’s Fourier factorization rules; DM, differential methods; RCWM, rigorous coupled-wave methods) for a sinusoidal deep grating made of anisotropic and conducting material. The steps with equal thickness, Δy=h/100, are used for both the differential and the rigorous coupled-wave methods, and the numerical integrations for the differential methods are given by the classical fourth-order Runge–Kutta scheme. (a) TE diffraction waves. (b) TM diffraction waves.

Fig. 5
Fig. 5

Same as Fig. 4, except for the step thickness of Δy=h/30. (a) TE diffraction waves. (b) TM diffraction waves.  

Fig. 6
Fig. 6

Convergences of diffraction efficiencies computed by using various integration schemes (DM, differential method; RCWM, rigorous-coupled wave method; ERK, classical fourth-order Runge–Kutta scheme; EAS, predictor–corrector Adams scheme; IMS, implicit midpoint scheme; IRK, implicit fourth-order Runge–Kutta scheme; IAS, implicit Adams–Moulton scheme). The same grating as in Fig. 4 is calculated for truncation orders (a) N=10 and (b) N=25.

Equations (38)

Equations on this page are rendered with MathJax. Learn more.

×E=ik0B,
×H=-ik0D,
Ex(x, y, z)=n=-NNEx,n(y)exp(ik0(αnx+γz)),
αn=1μ1sin θ cos ϕ+n λ0d,
γ=1μ1sin θ sin ϕ,
ddyf(y)=ik0M(y)f(y),
f(y)=[Ex][Ez][Hx][Hz].
f(y)=P(c)a(c)(y)=P11(c)P12(c)P21(c)P22(c)a(c)(y),
f(y)=P11(s)P21(s)a-(s)(y),
a(c)(y)=a-(c)(y)a+(c)(y),
a+(c)(yν)a-(s)(0)=Sν,11Sν,21a-(c)(yν),
a(c)(yν)=T(ν)α(c)(yν-1)=T11(ν)T12(ν)T21(ν)T22(ν)a(c)(yν-1),
Sν,11=(T21(ν)+T22(ν)Sν-1,11)(T11(ν)+T12(ν)Sν-1,11)-1,
Sν,21=Sν-1,21(T11(ν)+T12(ν)Sν-1,11)-1.
S0,11=-(P21(s)P11(s)-1P12(c)-P22(c))-1×(P21(s)P11(s)-1P11(c)-P21(c)),
S0,21=(P22(c)P12(c)-1P11(s)-P21(s))-1(×P22(c)P12(c)-1P11(c)-P21(c)).
T(ν)=P(c)-1F(ν,L)F(ν,L-1) . . . F(ν,1)P(c),
F(ν,μ)=P(ν,μ)U(ν,μ)P(ν,μ)-1,
P(ν,μ)=(p1(ν,μ) p2(ν,μ). . .p8N+4(ν,μ)),
(U(ν,μ))n,m=δn,mexp(ik0βn(ν,μ)Δy),
fμ(ν)=F(ν,μ)fμ-1(ν),
F(ν,μ)=I+16Rμ,1(ν)+13Rμ,2(ν)+13Rμ,3(ν)+16Rμ,4(ν),
Rμ,1(ν)=ik0ΔyMμ-1(ν),
Rμ,2(ν)=ik0ΔyMμ-1/2(ν)(I+Rμ,1(ν)/2),
Rμ,3(ν)=ik0ΔyMμ-1/2(ν)(I+Rμ,2(ν)/2),
Rμ,4(ν)=ik0ΔyMμ(ν)(I+Rμ,3(ν)).
f˜μ(ν)=fμ-1(ν)+ik0Δy24 (55Mμ-1(ν)fμ-1(ν)-59Mμ-2(ν)fμ-2(ν)+37Mμ-3(ν)fμ-3(ν)-9Mμ-4(ν)fμ-4(ν)),
fμ(ν)=fμ-1(ν)+ik0Δy24 (9Mμ(ν)f˜μ(ν)+19Mμ-1(ν)fμ-1(ν)-5Mμ-2(ν)fμ-2(ν)+Mμ-3(ν)fμ-3(ν)).
fμ(ν)-fμ-1(ν)Δy=ik0Mμ-1/2(ν)fμ(ν)+fμ-1(ν)2,
F(ν,μ)=I-ik0Δy2Mμ-1/2(ν)-1I+ik0Δy2Mμ-1/2(ν).
F(ν,μ)=I+12Δy(R1+R2),
R1=ik0Mμ-(3-3)/6(ν)I+14 ΔyR1+3+2312 ΔyR2,
R2=ik0Mμ-(3+3)/6(ν)I+3-2312 ΔyR1+14 ΔyR2.
R1=K1+148 Δy2K2-1-1I+3+2312 ΔyK2-1,
R2=K2-1I+3-2312 ΔyR1,
K1={ik0Mμ-(3-3)/6(ν)}-1-14 ΔyI,
K2={ik0Mμ-(3+3)/6(ν)}-1-14 ΔyI.
fμ(ν)=I-i9k0Δy24Mμ(ν)-1fμ-1(ν)+ik0Δy24×(19Mμ-1(ν)fμ-1(ν)-5Mμ-2(ν)fμ-2(ν)+Mμ-3(ν)fμ-3(ν)).

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